Nonautonomous impulsive systems with unbounded nonlinear terms

In Fenner and Pinto (1999) [23], proved that if the linear impulsive system satisfies the IS condition (see Definition 2.1) with bounded nonlinear term f(t,x,η)f(t,x,η), then the perturbed nonlinear impulsive system has a unique bounded solution (see Theorem A). The method used to prove Theorem A cannot be applied to the unbounded case. In this paper, we prove that if |f(t,x,η)|⩽μeβ|t|+M|f(t,x,η)|⩽μeβ|t|+M, then the perturbed nonlinear impulsive system has a unique solution.